A sum of noetherian modules is a noetherian module

Question

Assume $M=N_1+N_2$ is a module, where $N_1,N_2$ are noetherian modules. How can I show that $M$ is also noetherian?

Answer 1

The standard way to prove it is considering the exact secuence: $$0\longrightarrow N_1 \overset{i}\longrightarrow N_1\oplus N_2\overset{j}\longrightarrow N_2\longrightarrow 0$$ Where $i(n)=(n,0)$ and $j(n,m)=m$. As it is an exact secuence, the module in the middle, ie $N_1\oplus N_2$, is noetherian iff the two other modules are.

Answer 2

I think the sum need not be direct.However we can show $N_1 +N_2$ is isomorphic to a quotient of the direct sum of them.

Answer 3

First you can think about how the submodules of $N_1 \oplus N_2$ look like. Then you can use that a module is noetherian iff all of its submodules are finitely generated.